Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}x+5y &= -9 \\ 2x-9y &= 1\end{align*}$
Solution: Begin by moving the $x$ -term in the second equation to the right side of the equation. $-9y = -2x+1$ Divide both sides by $-9$ to isolate $y$ $y = {\dfrac{2}{9}x - \dfrac{1}{9}}$ Substitute this expression for $y$ in the first equation. $x+5({\dfrac{2}{9}x - \dfrac{1}{9}}) = -9$ $x + \dfrac{10}{9}x - \dfrac{5}{9} = -9$ Simplify by combining terms, then solve for $x$ $\dfrac{19}{9}x - \dfrac{5}{9} = -9$ $\dfrac{19}{9}x = -\dfrac{76}{9}$ $x = -4$ Substitute $-4$ for $x$ back into the top equation. $ -4+5y = -9$ $-4+5y = -9$ $5y = -5$ The solution is $\enspace x = -4, \enspace y = -1$.